Place halved baby potatoes in a medium pot, cover with cold salted water, bring to a boil and simmer 6–8 minutes until just tender; drain and toss with 1 teaspoon olive oil and a pinch of salt.

Spread 1 tablespoon butter in the bottom of a 6-quart slow cooker and arrange the par-cooked potatoes in an even layer; season chicken breasts with salt, pepper, and 1 teaspoon dried Italian seasoning and place over the potatoes.

Whisk together 1 1/2 cups chicken broth, 1/2 cup heavy cream, 3 tablespoons melted butter, 3 cloves minced garlic, and 1/2 cup of the grated Parmesan; pour evenly over the chicken and potatoes.

Cover and cook on LOW for 3 1/2–4 hours or on HIGH for 2–2 1/2 hours until chicken registers 165°F at the thickest point.

Remove chicken to a cutting board and tent with foil to rest 5–10 minutes; stir the crockpot sauce, temper 1 tablespoon cornstarch mixed with 2 tablespoons cold water by whisking a small amount of hot liquid into it, then return the tempered slurry to the crockpot and let sit 3–5 minutes to thicken.

Slice the chicken, stir in the remaining 1/4 cup Parmesan and optional lemon juice, taste for seasoning, and serve with potatoes spooned with sauce and a sprinkle of chopped parsley if desired.